Vertebral Disc Prosthesis

ABSTRACT

A prosthesis for a vertebral column has an upper part ( 10 ) for attachment to an upper vertebrae, a lower part ( 12 ) for attachment to a lower vertebrae and a middle part ( 11 ) located between the upper and the lower parts, wherein the upper part has a lower surface portion with a first radius of curvature, the middle part has an upper surface portion with a second radius of curvature and a lower surface with a third radius of curvature and the lower part has an upper surface with a fourth radius of curvature. The centre of the radius of curvature for at least two surfaces is offset rearwardly with respect to a central vertical axis ( 13 ) through the upper and lower vertebrae and/or the upper and lower parts. Also defined is device for linking bones, in the form of a band with attachment portions having a number of filaments that provide zones conducive to cellular growth as well as a method of modelling a prosthesis and a process for analysing performance of a prosthesis.

FIELD OF THE INVENTION

The present invention relates to a prosthesis primarily for use as anartificial invertebral disk, predominantly, but not exclusively, for usein human spines.

BACKGROUND OF THE INVENTION

A human invertebral disk maintains a linkage between adjacent vertebraeof the vertebral column. It must fulfil a number of important functionsincluding load bearing and dampening of impact forces. Furthermore, itmust permit a complex pattern of movements and resist various stresses,pure or combined, in the sagittal, coronal and axial planes. Assisted bymusco-ligamentous structures surrounding the spine, the invertebral diskmust also help to maintain the normal alignment of the vertebrae of thespinal column.

An ideal artificial disk replacement will accurately reproduce all thefunctions of the invertebral disk. However although there have been manydifferent artificial disks which have been described and tested, at thistime they have all failed to reproduce the abilities of an invertebraldisk.

Typical failings of previous artificial disks have included loosening ordislodgement of vertebral fixation, premature materials wear orstructural failure, poor replication of normal or physiological spinalsegmental motion and predisposition to the loss of normal neutralvertebral alignment.

An important aspect of the normal motion of the spinal column and thekinematics of the various invertebral motion segments is the behaviourof the motion segments during flexion and extension movements in thesagittal plane. Fundamental to the kinematics is the location of theinstantaneous axis of rotation (IAR). The IAR varies from level to levelwithin the spinal column and throughout flexion and extension movementsfor any given motion segment (level).

One type of spinal disk prosthesis is described in U.S. Pat. No.5,674,296. The endoprosthesis described consists of a resilient bodyhaving a generally elliptical shape. The endoprosthesis is affixedbetween adjacent upper and lower vertebrae through L-shaped supportseach having confronting concave-convex legs for engaging the adjacentbone sectional thickness on one surface and retaining the resilientendoprosthesis therebetween. The endoprosthesis is centrally locatedbetween the upper and lower vertebrae to allow central pivoting of theupper vertebrae relative to the lower vertebrae.

In addition to the above a gasket and seal are located at the anteriorand posterior regions between the vertebrae to seal the endoprosthesisin its position between the upper and lower vertebrae.

U.S. Pat. No. 5,556,431 describes another type of invertebral diskendoprosthesis in which top and bottom plates are used instead of theL-shaped supports of the above identified US patent. The endoprosthesisdescribed includes a core which has spherical upper and lower surfaceswhich from drawings shown appear to be aligned with a central verticalaxis through the upper and lower vertebrae.

In contrast to U.S. Pat. No. 5,674,296 the prosthesis core of thispatent has an edge rim which limits the range of movement of the coreand ensures even under extreme conditions cohesion of the prosthesis.

This patent also discloses displacement of the centre of articulation ofthe prosthesis towards the rear relative to the centre of the vertebralend plates so as to provide sufficient space in the ventral edge area ofthe prosthesis upper and lower plates so as to enable receipt of bonescrews.

Other artificial prostheses have sought to reproduce normal variation inthe location of the IAR using various mechanisms including the use ofvisco-elastic deformable cores. An example of this is shown in U.S. Pat.No. 5,824,094. Unfortunately these type of artificial disks are subjectto premature materials wear and stress failure. Furthermore, artificialdisks with metallic springs have not yet found their way into clinicaluse.

All of the artificial disks described above have inherent problems whichultimately create unnatural stresses and resultant pain for anartificial disk implant recipient. The present invention provides analternative prosthesis which is aimed at mitigating at least some of theproblems associated with prior art prosthesis.

SUMMARY OF THE INVENTION

It should be noted that definitions for abbreviations are provided atthe beginning of the details description of the drawings.

According to one embodiment of the present invention there is provided avertebral disk prosthesis which reproduces substantially similarkinematics of a human invertebral disk.

According to another embodiment of the invention a process for analysingprosthesis performance is provided using a unique modelling method todescribe motion of an artificial disk with a mobile core.

According to a further embodiment of the present invention the processof analysis involves a combination of linear algebra and matrixtransformations.

It is preferred that the process of analysis enables optimum design ofan invertebral disk endoprosthesis.

According to another embodiment of the present invention a prosthesisfor a vertebral disk is provided with a mobile core in which the axis ofrotation is able to vary, but which can more closely approximate thenormal anatomical centre of rotation (ACR) of an existing prosthesiswith a mobile core.

According to another embodiment of the invention a disk prosthesis isprovided which minimises the adverse effects of abnormal tension inadjacent ligamentis structures.

According to an object of one embodiment of the invention there isprovided a disk prosthesis which resists a tendency to adopt an abnormalposition or orientation at rest.

It is preferred that a prosthetic disk is provided which has a long lifeexpectancy.

According to one aspect of the present invention there is provided aprosthesis for a vertebral column comprising an upper part forattachment to an upper vertebrae, a lower part for attachment to a lowervertebrae and a middle part located between the upper and lower parts,wherein the upper part has a lower surface portion with a first radiusof curvature, the middle part has an upper surface portion with a secondradius of curvature and a lower surface portion with a third radius ofcurvature and the lower part has an upper surface portion with a fourthradius of curvature, wherein the centre of the radius of curvature forat least two surfaces is offset rearwardly with respect to a centralvertical axis through the upper and lower vertebrae.

Preferably the centre of the fourth radius of curvature and/or the firstradius of curvature is offset rearwardly of the central vertical axis.

It is preferred that the centre of the radius of curvature of all of thesurfaces is offset rearwardly with respect to the central vertical axis.

The centre of the radius of curvature for each of the surfaces ispreferably located in the posterior third of the prosthesis.

The middle part may have a minor central axis and a major central axis,the minor central axis being located through the centre of the radius ofcurvature of the second and third surfaces.

The minor central axis may be inclined with respect to the verticalcentral axis.

It is preferred that the major axis is located through the centre of theposterior and anterior ends of the middle part.

The second and third surfaces may have a substantially similar radius ofcurvature.

At least one of the second and third surfaces may have one of a convex,concave, cylindrical surface.

The posterior and anterior ends may comprise flat surfaces.

Preferably the middle part has a convex upper surface and a concavelower surface.

Preferably the upper surface of the middle part is concave and the lowersurface of the middle part is concave.

Preferably the radius of curvature of the upper surface of the middlepart is greater than the radius of curvature of the lower surface.

The flat surfaces may be vertically oriented or slightly skewed inaccordance with normal angulation of vertebrae.

It is preferred that the flat surfaces are vertically oriented parallelto the vertical axis plus or minus an angular offset.

According to one embodiment the flat surfaces are parallel to the minoraxis.

It is preferred that the centre of the radius of curvature for the thirdsurface is offset rearwardly with respect to the centre of the radius ofcurvature for the second surface.

The radius of curvature of the third surface according to one embodimenthas a centre on a line perpendicular to the major axis.

According to another embodiment the radius of curvature of the thirdsurface has a centre on a line coincident with the minor axis.

According to a further embodiment the radius of curvature of the secondsurface has a centre on a line at right angles/normal to the major axis.

According to a further embodiment the second surface has a radius ofcurvature with a centre on a line coincident with the minor axis.

According to a further embodiment the first and fourth surfaces haveradii of curvature with a centre similar to that for the third andsecond surfaces respectively.

It is preferred that the centre of the radius of curvature of the secondand/or third surfaces is substantially coincident with a vertical axisthrough the anatomical centre of rotation.

The length of the second and third surfaces may be substantially thesame.

Preferably the length of the end surfaces of the posterior and anteriorends is different.

The posterior end surface may be larger than the anterior end surface ifthe second and third surfaces are convex.

Preferably if the second and third surfaces are concave then theposterior end surface is smaller than the anterior end surface.

According to one embodiment the second surface has a major portionlocated forward of the anatomical centre of rotation.

The third surface may have a major portion located forward of theanatomical centre of rotation.

It is preferred that each of the surfaces have a major portion locatedforward of the anatomical centre of rotation and the minor portionlocated rearwardly of it.

The middle part may be asymmetric.

Preferably a major portion of the middle part is located forward of theanatomical centre of rotation when the upper and lower vertebrae aresubstantially vertically aligned.

According to one embodiment the minor axis of the middle part when in avertical orientation close to its point of rest (equilibrium with theupper and lower vertebrae) is as close as possible if not coincidentwith a vertical axis through the anatomical centre of rotation.

The upper part may comprise an axis of symmetry which is offset to theposterior end.

The axis of symmetry may coincide with the centre of radius of curvatureof the first surface.

The axis of symmetry preferably passes through the anatomical centre ofrotation.

The lower part may comprise an axis of symmetry which passes through theanatomical centre of rotation.

Preferably the first and second surfaces have substantially matchingradii of curvature.

Preferably the third and fourth surfaces have substantially matchingradii of curvature.

The upper part may comprise an anterior portion which is larger than aposterior portion relative to the axis of symmetry.

The lower part may comprise an anterior portion which is larger than aposterior portion relative to the axis of symmetry.

It is preferred that the middle part is movable relative to the upperand lower parts.

Movement of the middle part is preferably limited by stopping meanslocated behind and in front of the middle part.

The stopping means may include end portions of the upper and lowerparts.

The upper and lower parts may be fixed to the upper and lower vertebraeand configured to form a small gap between respective anterior endportions and a larger gap between respective posterior end portions.

Preferably the second and/or third surfaces include a curved surfaceportion.

The curved surface portion preferably has a substantially sphericalprofile with a radius of curvature.

It is preferred that the second and third surfaces have centres ofradius of curvature which are vertically offset.

Preferably the first and second surfaces have substantially similarradii of curvature of opposite sign.

The third and fourth surfaces may have substantially similar radii ofcurvature of opposite sine.

According to one embodiment the second radius of curvature is differentthan the third radius of curvature.

According to an alternative embodiment the third radius of curvature isgreater than the first or less than the first.

The third surface may be offset more than the second from the centralvertical axis of the vertebrae.

It is preferred according to one embodiment that the parts of theprosthesis are designed asymmetrically to correspond to the asymmetry ofupper and lower vertebrae with which they are to be used.

It is to be understood that any of the embodiments or preferred optionsdescribed previously include variations in which all surfaces are tiltedor skewed.

It is preferred that the lower part and upper part include a stopsurface at a rearward part to limit rearward movement of the middlepart.

The length of one of the second/third surfaces may be greater than theother when measured front to back. The fourth surface preferablyincludes a flat forward portion extending from a front end of a curvedportion.

The curved portion preferably has a spherical cylindrical profile.

It is preferred that the top and bottom surfaces are convex.

According to another aspect of the present invention there is provided adevice for linking bones comprising a band having first and second endseach with attachment portions for attachment to upper and lower bonesand a plurality of filaments configured to provide a plurality of zonesconducive to cellular growth.

It is preferred that the plurality of zones comprise spaces.

The plurality of filaments may be configured to form a matrix.

According to one embodiment the plurality of zones comprise a pluralityof interwoven portions.

The filaments may be woven together.

The band preferably comprises a gauze or mesh.

The band may have inherent stiffness.

Preferably the band is resiliently deformable.

It is preferred that the band is extendible and compressible.

The zones may comprise spaces between filaments.

The zones according to one embodiment include overlapping regions offilaments.

Preferably the spaces are formed by filaments.

According to another embodiment the filaments are configured in paralleland perpendicular rows forming an intersecting grid pattern.

It is preferred that the device is used for linking upper and lowervertebrae.

It is preferred that the band is connected to an anterior portion ofupper and lower vertebrae.

The band may be generally flat.

The band may be in the form of a flat strap.

The band may be composed of fabric, metal or a polymeric substance.

It is preferred that the band is made from a substance which dissolvesin use.

The band preferably can concertina or lozenge.

According to one embodiment the band provides axial support against apredetermined level of compression.

According to a further embodiment the band provides a predeterminedlevel of resilient extension.

Each attachment portion may comprise a plate or strap with holes toallow fixing elements to be inserted therethrough.

According to another aspect of the present invention there is provided aprosthesis for vertebrae having one or more of the features of thepreviously described prosthesis wherein the upper part when theprosthesis is attached to upper and lower vertebrae, closely simulatesrotational and translational movements possible with an invertebraldisk.

According to another aspect of the present invention there is provided amethod of producing a prosthesis for vertebrae comprising providing amodel for designing a prosthesis used to simulate kinematics of aninvertebral disk, using the model to produce a prosthesis comprising anupper part, a lower part and a middle part, which prosthesis simulateskinematics of an invertebral disk and wherein the upper part when theprosthesis is attached to upper and lower vertebrae simulates rotationaland translational movements possible with an invertebral disk.

Preferably the simulation provided by the prosthesis includes tilting ofthe upper part relative to the anatomical centre of rotation of thelower vertebral disk.

The simulation provided by the prosthesis may include movement duringrotation along an arc permissible with an invertebral disk.

The simulation provided by the prosthesis may include translationalmovement forward and back to an extent permissible for an uppervertebrae with an invertebral disk.

It is to be noted that the anatomical centre of rotation may vary foradjacent pairs of upper and lower vertebrae in a vertebral column.

According to one embodiment the radius of curvature for the first andsecond surfaces is selected based on rotational movement possible for anupper vertebrae with respect to a lower vertebrae.

According to another embodiment the third and fourth surfaces have aradius of curvature which is selected to simulate the amount of tiltingpossible for the upper vertebrae.

It is preferred that the angle of tilting permissible for the uppervertebrae and the angle indicative of the rotational movement of theupper vertebrae together closely approximate the angular displacement ofan upper vertebrae with respect to a lower vertebrae with an invertebraldisk between the upper and lower vertebrae.

According to another aspect of the present invention there is provided aprocess for analysing performance of a prosthesis for use between upperand lower vertebrae, the process comprising determining a first centreof radius of curvature for a lower surface of a middle part of aprosthesis, determining a second centre of radius of curvature for anupper surface of the middle part of the prosthesis, providing a linkbetween the first centre of radius of curvature and second centre ofradius of curvature, rotating the second centre of radius of curvaturewith respect to the first centre of radius of curvature by a degreesrepresenting tilting of the upper vertebrae, rotating a portion of thefirst link by β degrees whereby the length of the portion corresponds tothe length from the second centre of rotation of curvature to the centreof the lower surface of the upper vertebrae or upper surface of theupper part whereby β corresponds to angular movement of the upper partover the upper surface of the middle part, determining the anatomicalcentre of rotation, determining an angle γ corresponding to the desiredangle of rotation of an invertebral disk relative to the anatomicalcentre of rotation, comparing the angleγ with the angles α+β anddesigning a prosthesis with values for the upper and lower centre ofradius of curvature which minimises the value of γ−(α+β).

According to a further aspect of the present invention there is provideda process similar to the process described above except that the firsttwo determining steps are replaced by the steps of determining an uppercentre of radius of curvature for an upper surface of a lower part of aprosthesis and determining a lower centre of radius of curvature for alower surface of an upper part of the prosthesis.

According to one embodiment the link passes through the minor axis ofthe middle part.

According to another embodiment the angle α corresponds to the anglebetween the upper centre of radius of curvature relative to a centralvertical axis of the upper and lower vertebrae (prosthesis axis).

According to one embodiment the angle β corresponds to the angle formedby moving the first link through an angle whereby the link coincideswith a central point on the lower surface of the upper vertebrae whenmoved a maximum permissible amount relative to the anatomical centre ofrotation.

According to different embodiments of the present invention the secondand third surfaces may be any one of the following combinations:

convex/convex;

concave/concave;

concave/convex;

convex/concave;

convex/cylindrical;

concave/cylindrical.

It is preferred that the process includes determining the length of thefirst link and the length of a second link between the lower centre ofradius of curvature and the centre point on the lower surface of theupper vertebrae.

According to a further embodiment of the present invention the methodinvolves converting a frame located at the anatomical centre of rotationto a global co-ordinate system and moving the frame by translational androtational transformations to relocate the frame at either the centre ofthe lower surface of the upper vertebrae or a point on the lower surfaceof the upper vertebrae that lies on a vertical axis through theanatomical centre of rotation when the upper vertebrae is in rest abovethe lower vertebrae.

It is preferred that the transformations involved include the algebraicand matrix transformations described in the preferred embodiment.

According to one embodiment the process involves designing theprosthesis so that the maximal change in ligament length due toprosthesis malplacement is minimised. Prosthesis malplacement can bedefined by the value of the horizontal distance between the prosthesisaxis and the patients centre of rotation (value Ldsk in FIGS. 5 a, 5 cand value L in FIGS. 19A and 19B).

According to another embodiment the process involves designing amechanism such that the ligament is stretched in such a way as to beunder more tension in flexion and extension and be under the leasttension in the neutral position. Such a mechanism will provide arestoring force that will tend to move the prosthesis back to a neutralposition.

According to a further aspect of the present invention there is provideda modelling method for a prosthesis comprising:

determining a frame matrix FR1 in at least 2D for a prosthesis in situbetween upper and lower vertebrae representing a co-ordinate system fora reference point at the ACR of a linear vertebrae.

determining a reference frame B1for a point at the CUPR expressed interms relative to the frame FR1 at the ACR;

where ${B\; 1} = \begin{bmatrix}1 & 0 & l \\0 & 1 & p \\0 & 0 & 1\end{bmatrix}$ where

l=the distance of the CUPR from the ACR along an x axis; orp=the distance of the CUPR from the ACR along a y axis.

Rotating the frame B1 by α° to produce a new frame B2=B1×T

where α is the angle of rotation of the CLPR in relation to the CUPR;and

T is a transformation matrix:

$\lbrack  \quad \begin{matrix}{\cos \; \alpha} & {{- \sin}\; \alpha} & {\Delta \; x} \\{\sin \; \alpha} & {\cos \; \alpha} & {\Delta \; y} \\0 & 0 & 1\end{matrix} \rbrack $

translating the frame B2 by the distance b of the CUPR to the CLPR alongthe y-axes to produce a frame B3:

${{where}\mspace{14mu} {the}\mspace{14mu} {translation}\mspace{14mu} {matrix}} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & {- b} \\0 & 0 & 1\end{bmatrix}$

rotating the frame B3 by β degrees using T to produce a new frame

B4=B3×T

where β is the angle of rotation of a point B on an upper vertebraerelative to the CLPR,

translating the frame B4 by the distance C of the CLPR to point B alongthe y axes to produce a new frame B5.

where ${B\; 5} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & C \\0 & 0 & 1\end{bmatrix}$

translating the frame B5 by the distance l of the point B along thex-axis to a point E co-axial with a vertical axis through the ACR, toproduce a new frame B6

where the translation matrix is

$\lbrack  \quad \begin{matrix}1 & 0 & l \\0 & 1 & 0 \\0 & 0 & 1\end{matrix} \rbrack $

rotating frame A1 by γ using T to produce a new frame A2.

γ=normal rotation of an upper vertebrae relating to the ACR.

Translating A2 by a distance h of the ACR to point E along the y-axis toproduce a frame A3 where the translation matrix is

$\lbrack  \quad \begin{matrix}1 & 0 & 0 \\0 & 1 & h \\0 & 0 & 1\end{matrix} \rbrack $

Comparing B6 and A3 to determine how clearly the prosthesis simulateskinematics of an invertebral disk.

According to one embodiment frames B6 and A3 are rotated by γ° aboutglobal reference frame A1 to produce new frames A4 and B7.

Preferably the step of comparing includes solving at least one of thefollowing equations for a minimum value.

A3(1,3)−B6(1,3)=0 A4(1,3)−B7(1,3)−0

A3(2,3)−B6(2,3)=0 or A4(2,3)−B7(2,3)=0

where the numbers in brackets represent rows and columns respectively ofthe applicable matrix.

According to another embodiment the step of comparing includes solvingsimultaneous equations for equivalent rows and columns of A4 and B7.

It is preferred that reference frame A1 is a global reference frame.

It is to be understood that use of the word simulation is intended to beinterpreted broadly to cover similar and not just exact reproductions.

The word “prosthesis” is intended to cover any artificial insert havingany number of components.

The modelling method used for analysing performance of a prosthesispreferably describes motion of an artificial disk that has a mobile coreand is constrained by adjacent ligamentous structures.

The modelling method preferably can be used to optimise the variousdesign parameters of a mobile core prosthesis so as to more accuratelyreproduce the location of the IAR of a normal disk and minimise thetendency to follow or adopt an abnormal path of motion duringflexion/extension movements and/or an abnormal neutral alignment in thesagittal plane at rest.

Using the modelling method it can be shown that for a prosthesis with amobile core possessing upper and lower plates with articulatingsurfaces, according to a preferred embodiment of the present inventionthe following applies:

1. The larger the radius, the more the core will need to translate for agiven change in orientation.

2. The smaller the radius, the less the core will need to translate fora given change in orientation.

3. For a given change in position and orientation:

-   -   (a) The closer the axis of rotation of the prosthesis is to the        normal anatomical centre of rotation, the less the LLS need to        change length.    -   (c) The more the radii of the upper and lower articulating        surfaces of a bi-convex or bi-concave prosthesis are unequal,        the more the LLS need to stretch, if the axis of rotation of the        prosthesis is displaced anterior to the anatomical centre of        rotation.    -   (d) If the axis of rotation of the disc prosthesis is displaced        anterior to the normal anatomical axis of rotation, during        flexion, the final position and orientation of the upper        vertebra will be determined by the ability of both the PLL and        the LLS to stretch. It follows that there four possibilities:    -   (i) PLL can't stretch & LLS can't stretch—the upper vertebra        cannot move,    -   (ii) PLL can stretch & LLS can't stretch—the upper vertebra will        adopt a position of kyphosis,    -   (iii) PLL can stretch and LLS can stretch—the upper vertebra        will be unstable and may adopt a non-anatomical        position/orientation    -   (iv) PLL can't stretch and LLS can stretch —unlikely to occur in        clinical practice

It follows that for the upper vertebra to adopt a given orientationduring flexion, the LLS must stretch and therefore the final vertebralposition will not be normal.

-   -   (e) If the axis of rotation of the disc prosthesis is displaced        anterior to the normal anatomical axis of rotation, during        extension, the final position and orientation of the upper        vertebra will be entirely determined by the ability of the LLS        to stretch. This is because the ALL has been resected during the        surgical approach. It follows that there are two possibilities:    -   i) The LLS can't stretch—the upper vertebra will adopt a        position of less lordosis than normal.    -   (ii) The LLS can stretch—the upper vertebra can adopt the normal        orientation but will have an abnormal position which is        permitted by stretch of the LLS.

4. Movement of the prosthesis axis of rotation close to the normal ACRwill:

-   -   (i) Minimize the need for the ligaments to stretch or shorten        during normal flexion and extension movements    -   (ii) Optimize the ability of the vertebra to adopt normal        orientation and position during flexion and extension movements

5. Movement of the prosthesis axis towards the normal anatomicalposition for the ACR lying below the posterior half of the disc spaceintroduces two new problems:

-   -   (i) Posterior translation of the core on flexion, with an        existing bi-convex design, causing neural compression. A        solution is to use a bi-concave core. A bi-concave core        mechanism will cause the core to move anteriorly with flexion        and posteriorly with extension.    -   (ii) In some embodiments the core becomes asymmetrical around        the prosthetic axis. Rotation around this axis would therefore        produce neural compression. One solution to prevent rotation        around the prosthetic axis is by making one of the two        prosthetic articulations cylindrical rather than spherical. A        further solution is to make one of the two prosthetic        articulations an ellipsoid shape. Yet another solution is to        have both surfaces spherical but placing mechanical stops or        guide fins.

6. In another embodiment, the Mathematical Process can be used tooptimize a disc mechanism consisting of curved upper and lowerarticulations where the arc centres are below the disc base but wherethe radii are unequal. This may permit variation in the verticallocation of the prosthesis axis of rotation but restrict it to below thedisc base. Such a prosthesis would not have the ability to achievecertain undesirable positions that would be readily apparent to someoneskilled in the art.

7. It follows that following resection of the anatomical ALL foranterior insertion of an internally unconstrained disc prosthesis, thatthe prosthesis may not function correctly without appropriate tension inthe adjacent ligamentous structures. The placement of constraints withinthe disc prosthesis will strain the prosthesis/vertebral interface andmay predispose to loosening of the prosthesis. However under somecircumstances it may be desirable to allow the placement of materialthat is attached to the lower non articulating surface of the upper partand to the upper non articulating surface of the lower part. Suchmaterial could be made out of any appropriate elastic material (such as,but not restricted to, a polymer) that could increase the stiffness ofthe construct in a desirable way. While the Mathematical Process may beused to design a prosthesis which will minimize the effect of abnormaltension in the adjacent ligamentous structures, the prosthesis mayoptimally be further supported by the placement of an artificial ALL,attached to anterior aspect of the vertebral bodies and separate fromthe disc prosthesis.

It is preferred that following resection of the anatomical LLL foranterior insertion of an internally unconstrained disk prosthesis, thatthe prosthesis may not function correctly without appropriate tension inthe adjacent ligamentous structures. The placement of constraints withinthe disk prosthesis will strain the prosthesis/vertebral interface andmay predispose to loosening of the prosthesis. While the mathematicalprocess may be used to design a prosthesis which will minimise theeffect of abnormal tension in the adjacent ligamentous structures, theprosthesis may optimally be further supported by the placement of anartificial ALL, attached to anterior aspect of the vertebral bodies andseparate from the disk prosthesis.

In the claims which follow and in the preceding description of theinvention, except where the context requires otherwise due to expresslanguage or necessary implication, the word “comprise” or variationssuch as “comprises” or “comprising” is used in an inclusive sense, i.e.to specify the presence of the stated features but not to preclude thepresence or addition of further features in various embodiments of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments of the present invention will now be described byway of example only with reference to the accompanying drawings inwhich:

FIG. 1 shows a schematic diagram of a prior art prosthesis between upperand lower vertebrae;

FIG. 2 shows a dual linkage model of a prosthesis in accordance with anembodiment of the present invention;

FIG. 3 shows a schematic of motion of a normal invertebral disk about ananatomical centre of rotation;

FIG. 4 shows a schematic diagram of upper and lower vertebrae withattached global reference frame in accordance with a preferredembodiment of the present invention;

FIGS. 5A and 5C show a schematic diagram of a prosthesis (convex/concaveand biconcave core respectively) and upper and lower vertebrae showingtranslational characteristics of a model according to the preferredembodiment of the invention;

FIGS. 5B and 5D show rotational characteristics of the model shown inFIGS. 5A and 5C;

FIG. 6 shows a schematic of a bi-convex core prosthesis with an uppervertebrae in kyphosis;

FIG. 7 shows a schematic of a convex/concave core prosthesis with theupper vertebrae in kyphosis;

FIG. 8 shows a schematic diagram of a biconvex prosthesis with the uppervertebrae under the constraint of maximum ligament stretch (MLS);

FIG. 9 shows a schematic diagram of a prosthesis with a core having aconvex upper surface and concave lower surface, with the upper vertebraeunder the constraint of maximum ligament stretch (MLS);

FIG. 10A shows a prosthesis according to another embodiment with upperand lower vertebrae in rest positions;

FIG. 10B shows the prosthesis shown in FIG. 10A with the upper vertebraerotated by 10°;

FIG. 11 shows a prosthesis according to another embodiment of thepresent invention with the upper vertebrae and lower vertebrae at rest;

FIG. 12 shows the prosthesis shown in FIG. 11 with the upper vertebraerotated by 10°;

FIG. 13A shows a top view of a prosthesis according to anotherembodiment of the present invention;

FIG. 13B shows a cross-sectional view of the prosthesis of FIG. 13Ataken along sectional lines A-A;

FIG. 13C shows a cross-sectional view of the prosthesis shown in FIG.13A taken along sectional lines B-B;

FIG. 13D shows a top view of the prosthesis shown in FIG. 13A;

FIG. 13E shows a rear view of the prosthesis shown in FIG. 13A;

FIG. 13F shows a side view of the prosthesis shown in FIG. 13A with theleft hand side representing the posterior end;

FIG. 14 shows an angled view of a prosthesis according to anotherembodiment of the present invention;

FIG. 15A shows a side schematic view of a prosthesis according toanother embodiment of the invention with upper and lower vertebrae in arest position;

FIG. 15B shows the prosthesis in FIG. 15A with the upper vertebraerotated 10°;

FIG. 16 shows a schematic side view of a prosthesis according to anotherembodiment of the present invention;

FIG. 17 shows a schematic side view of a prosthesis according to afurther embodiment of the present invention; and

FIG. 18 shows a front view of a ligament band of the present inventionaccording to one embodiment;

FIG. 19A shows a schematic cross-sectional end view of a prosthesis inan equilibrium position according to another embodiment of theinvention;

FIG. 19B shows the prosthesis of FIG. 19A in an unstable position;

FIGS. 20A and 20B show a three dimensional graphical analysis ofdifferent positions of a prosthesis having a core with a convex uppersurface and convex lower surface in accordance with an embodiment of thepresent invention;

FIG. 21 shows a 2D graphical representation of a the prosthesis analysedin FIG. 20;

FIG. 22 shows a 3D graphical analysis of a bi-convex prosthesis;

FIG. 23 shows a 2D graph of ligament length vs angular movement for adual convex prosthesis; and

FIG. 24A and 24B show a 3D graphical analysis of a Bi-concave prosthesisaccording to different embodiments of the present invention.

DETAILED DESCRIPTION OF THE DRAWINGS

To assist with an understanding of the invention terminology used is setout below.

Terminology:

-   a. Centre of Rotation (COR): A point around which an object is    rotated to achieved a desired position and orientation with zero    translation. (Translation is defined as a pure linear movement in    any direction without change in orientation).-   b. Instantaneous Axis of Rotation (IAR): The location of the COR at    any instant in time as it varies in exact location during the course    of movement (such as flexion and extension) between two end points.-   c. Anatomical Centre of Rotation (ACR): The centre of rotation of an    undiseased cervical motion segment between two end points (such as    flexion and extension).-   d. Upper and lower Prosthesis Radii (UPR & LPR): The upper and lower    radii of curvature of the disc prosthesis.-   e. Centre of Upper and Lower Prosthesis Radii (CUPR & CLPR): The    centre point of the upper and lower disc prosthesis radii. For a    bi-convex disc prosthesis core, the CUPR lies inferior and the CLPR    lies superior.-   f. Prosthetic axis (PA): The line joining the CUPR and LUPR.-   g. Lateral ligament structure (LLS): The ligaments taking origin    from the supero-lateral edge of the lower vertebra and attached to    the infero-lateral edge of the upper vertebra, along lines radiating    upwards and forwards from the ACR and which stretch the least during    vertebral segmental flexion and extension around the ACR.-   h. Simplified lateral ligament structure (SLLS): A single line or    intervertebral linkage which describes the mathematical behaviour of    the LLS.-   i. Anterior longitudinal ligament (ALL): The anterior ligamentous    structures.-   j. Posterior longitudinal ligament (PLL): The posterior ligamentous    structures.-   k. The Mathematical Process: A mathematical process involving linear    algebra and matrix transformations which can be used to describe the    motion of an artificial disc that has a mobile core

FIG. 1 shows a prosthesis with a bi-convex core representing a prior artprosthesis as shown for example in U.S. Pat. No. 5,674,296 to Bryan.

From FIG. 1 it should be apparent that the upper vertebrae 10 can rotaterelative to the core 11 and the core 11 can rotate relative to the lowervertebrae 12.

It has been assumed in the past that because there is in effect twoangles of rotation, that the prosthesis can adopt whatever position isneeded to simulate normal rotation. However an analysis in accordancewith a preferred embodiment of the invention shows that exact simulationof normal rotation is not possible but it is possible to design aprosthesis with near normal motion.

Incremental normal rotation in the sagittal plane occurs around aninstantaneous centre of rotation. When measured over larger angles thisICR moves somewhat, although in both the lumber and cervical spines itis always in the posterior one half of the lower vertebrae.

In accordance with one embodiment of the invention, motion of the uppervertebrae 10 can be described by analysing it as a dual linkage withlinks 14 and 15 as shown in FIG. 2. Point CUPR remains fixed in globalco-ordinates. The motion can be considered as sequential movements ofthe links 14 and 15. Initially upper vertebrae 10, the core 11 and thepoint CLPR rotate by α degrees around the point CUPR. The lowervertebrae 12 then rotate by β degrees around the newly rotated positionof CLPR (CLPR¹).

The minor axis (not shown) of the core 11 remains at right angles tolink A which itself passes through the minor axis of the core 11. Core11 therefore moves in the same direction to upper vertebrae 10. Inflexion core 11 will anteriorly, in extension core 11 will moveposteriorly.

In designing a prosthesis as previously outlined it is desirable tosimulate as closely as possible movement of vertebrae in normaloperation with an invertebral disk between upper and lower vertebrae.Therefore to provide a frame of reference of this normal motionreference is made to FIG. 3 which shows motion of a normal disk,(invertebral disk) with the approximation of a fixed centre of rotation(ACR). All points on vertebrae 10 move to corresponding points onvertebrae 18 and the transformation that describes the movement of anyarbitrary point from the position of upper vertebrae 10 to uppervertebrae 16 is rotation by angle γ around ACR. Lines 17 and 18 bothexhibit positional information and angular information. Thesecharacteristics are defined as position and orientation.

It follows that for any artificial disk mechanism to reproduce thebehaviour of the movement shown in FIG. 3 that it must be able to moveline segment 17 to line segment 18 and at the end of the movement boththe position and orientation of line segment C1-D1 with the artificialdisk mechanism (prosthesis) must match the line segment 18 in FIG. 3.

Referring back to FIG. 2 it follows that the position and orientation ofthe vertebrae are fully described by angles α and β and the lengths ofthe links 14 and 15. It follows that if the mechanism in FIG. 2 is ableto mimic the mechanism in FIG. 3 (normal) then there must exist acombination of values for variables α,β,14,15 that will make both theposition and orientation of both vertebrae the same.

Position and orientation of objects in two dimensional space areconveniently describe by the use of linear algebra. To fully describethe position and orientation of a two dimensional structure in twodimensional space, a coordinate system can be attached to the object.This coordinate system is called a frame. All points on the movingobject have fixed coordinates in the new frame and the frame isconsidered to move within another coordinate system—usually the globalor ‘world’ coordinate system. FIG. 4 shows a Frame FR1 attached to themoving vertebrae in FIG. 3. The origin of this frame is displaced fromthe origin of the global frame G by position vector p. The orientationof frame FR1 is given by the unit vectors n for the x axis and o for they axis of FR1.

In matrix notation the frame FR1 can be described as

${{FR}\; 1} = \begin{bmatrix}{\overset{\_}{n}}_{x} & {\overset{\_}{o}}_{x} & {\overset{\_}{p}}_{x} \\{\overset{\_}{n}}_{y} & {\overset{\_}{o}}_{y} & {\overset{\_}{p}}_{y} \\0 & 0 & 1\end{bmatrix}$

Where n _(x)=x coordinate of unit vector n

n _(y)=y coordinate of unit vector n

ō_(x)=x coordinate of unit vector ō

ō_(y)=y coordinate of unit vector ō

p _(x)=x coordinate of position vector p

p _(y)=y coordinate of position vector p

Any point with coordinates x,y attached to frame FR1 can be converted toglobal coordinates by premultiplying matrix FR1 by the vector of thecoordinates of the point in FR1

${{{FR}\; 1} \star \begin{bmatrix}x \\y \\1\end{bmatrix}} = \begin{bmatrix}x_{global} \\y_{global} \\1\end{bmatrix}$

Any frame such as FR1 can be transformed by multiplying by atransformation matrix T with the following characteristics.

$T = \begin{bmatrix}{\cos \; \alpha} & {{- \sin}\; \alpha} & {\Delta \; x} \\{\sin \; \alpha} & {\cos \; \alpha} & {\Delta \; y} \\0 & 0 & 1\end{bmatrix}$

Where α=angle of rotation

Δx and Δy=change in x and y position.

If matrix M is premultiplied by Frame FR1 frame FR1 will be rotatedaround the fixed global reference frame origin and translated in thedirection of the global reference frames axes. If Matrix M ispostmultilpied by FR1, FR1 is rotated around the origin of the movingframe (FR1) and translated in the direction of the moving (FR1) framesaxes.

FIGS. 5A to 5D show a hypothetical prosthesis with a convex uppersurface and a concave lower surface. For analysis purposes there is amechanical linkage consisting of line segment AD rotating around point Aand a further link consisting of line segment DB. DB is rigidly attachedto the upper vertebrae and upper prosthetic end plate. A reference framehas been attached at point ACR. A further reference frame has beenattached at point A.

Considering the variables in FIGS. 5A and 5C it should be apparent thatBFR1 should have the following value—expressed in the global referenceframe AFR1.

${{BFR}\; 1} = \begin{bmatrix}1 & 0 & {Ldsk} \\0 & 1 & {Pdsk} \\0 & 0 & 1\end{bmatrix}$

In order for the reference frame BFR1 to be transformed to be attachedto the top vertebrae at point B, it must undergo the followingtransformations shown in FIGS. 5B and 5D.

1. Rotation by alpha degrees to produce new frame BFR1R

${{BFR}\; 1\; R} = {{{BFR}\; 1} \star \begin{bmatrix}{\cos \; \alpha} & {{- \sin}\; \alpha} & 0 \\{\sin \; \alpha} & {\cos \; \alpha} & 0 \\0 & 0 & 1\end{bmatrix}}$

in FIG. 1 alpha is negative considering the normal convention ofpositive rotation being anticlockwise.

2. Inferior translation by Bdsk in the frame of reference of BFR1R toproduce new frame BFR2

${{BFR}\; 2} = {{{BFR}\; 1R} \star \begin{bmatrix}1 & 0 & 0 \\0 & 1 & {- {Bdsk}} \\0 & 0 & 1\end{bmatrix}}$

3. Rotation of BFR2 by Beta degrees in its own frame of reference toproduce new frame BFR2R

${{BFR}\; 2R} = {{{BFR}\; 2} \star \begin{bmatrix}{\cos \; \beta} & {{- \sin}\; \beta} & 0 \\{\sin \; \beta} & {\cos \; \beta} & 0 \\0 & 0 & 1\end{bmatrix}}$

4. Translation by Cdsk in the frame of reference BFR2R to produce a newframe BFR3

${{BFR}\; 3} = {{{BFR}\; 2R} \star \begin{bmatrix}1 & 0 & 0 \\0 & 1 & {Cdsk} \\0 & 0 & 1\end{bmatrix}}$

BFR3 is now attached to the upper vertebrae at point B and has theorientation of the upper vertebrae. BFR3(1,3) (row 1, column 3) containsa function f(alpha,Beta) that represents the x coordinate of point B andBFR3(2,3) contains a function g(alpha, Beta) that represents the ycoordinate of point B. BFR3(1,1) contains a function k (alpha,Beta) thatrepresents the cosine of the angle made by the top vertebrae with theglobal reference frame.

Consider a further linear translation of −Ldsk in the frame of referenceof BFR3 (the upper vertebrae. This will create at new frame BFR4 atpoint E

${{BFR}\; 4} = {{{BFR}\; 3} \star \begin{bmatrix}1 & 0 & {- {Ldsk}} \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}$

The equivalent functions f, g and k now represent the coordinates ofpoint E and the (unchanged) angle of orientation of the upper vertebrae.

By performing the matrix calculations It can be shown that

f(α,β)=−(cos α·cos β−sin α·sin β)·Ldsk+(−cos α·sin β−sin α·cosβ)·Cdsk+sin α·Bdsk+Ldsk)  (1)

Where f=x coordinate of point E

g(α,β)=−(sin α·cos β+cos α·sin β)·Ldsk+(cos α·cos β−sin α·sinβ)·Cdsk−cos α·Bdsk+Pdsk)  (2)

Where g=y coordinate of point E And

k(α,β)=cos α·cos β−sin α·sin β  (3)

Where k=cosine of angle between upper vertebrae and global referenceframe.

From FIG. 5 it can be seen that as AFR1 is the global reference frameit's value is

${{AFR}\; 1} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$

a frame APFR2 can be derived by rotation by angle gamma (the desiredrotation of the normal disc) of frame AFR1 to produce AFR1R

${{AFR}\; 1R} = {{{AFR}\; 1} \star \begin{bmatrix}{\cos \; \gamma} & {{- \sin}\; \gamma} & 0 \\{\sin \; \gamma} & {\cos \; \gamma} & 0 \\0 & 0 & 1\end{bmatrix}}$

Frame AFR1R can now be translated by value Adsk along the y axis of AF1Rto produce frame AFR2

${{AFR}\; 2} = {{{AFR}\; 1R} \star \begin{bmatrix}1 & 0 & 0 \\0 & 1 & {Adsk} \\0 & 0 & 1\end{bmatrix}}$ ${{AFR}\; 2} = \begin{bmatrix}{\cos \; \gamma} & {{- \sin}\; \gamma} & {{- \sin}\; {\gamma \cdot {Adsk}}} \\{\sin \; \gamma} & {\cos \; \gamma} & {\cos \; {\gamma \cdot {Adsk}}} \\0 & 0 & 1\end{bmatrix}$

AFR2(1,3) should now contain the x coordinates of point E and AFR2(2,3)should now contain the y coordinates of point E.

Let function s(γ)=AFR2(1,3) (x coordinate)  (4)

Let Function t(γ)=AFR2(2,3) (y coordinate)  (5)

It follows that as both frame AFR2 and BFR4 are at the same point (E)that from equations 1 and 2 that

s(γ)=f(α,β)  (6)

t(γ)=g(α,β)  (7)

Equations 6 and 7 represent 2 simultaneous equations with two variables.In order for the mechanism to exactly simulate the movement of thenormal disc, it also follows the AFR2 and BFR4 must be equal.

AFR2=BFR4  (8)

It can be shown that for this to occur that as well as equations 6 and 7holding true. It also follows that.

λ=α+β  (9)

It can be shown by numerical means that there are no real solutions thatsatisfy equations 6, 7 and 8. For a given angle γ that a normal discwill flex, the solutions for angles α and β are such that the prosthesiswill be positioned in relative Kyphosis or Lordosis. FIG. 6 representsthe effect of an attempt to flex an existing prosthesis with a biconvexcore by 10 degrees with the constraint (constraint 1) being that point Eis the same as the normal prosthesis. The solution to equations 6, 7 and8 result in a equalling −10.72° and β equalling −18.26°. The dashed linerepresents a real disk rotating by 10° about .ACR, this positionrepresents the kyphotic solution to keep points E with the sameco-ordinates. This position is the position of 0 ligament stretch (ZLS).FIG. 7 represents the effect of an attempt to flex a prosthesis with acore with a convex uppersurface and a concave lower surface by 10degrees with the constraint (constraint 1) being that point E is thesame as the normal prosthesis. The solution to equations 6, 7 and 8result in a equalling 5.71° and β equalling −7.71°. The dashed linerepresents a real disk rotating by 10° about .ACR, this positionrepresents the kyphotic solution to keep points E with the sameco-ordinates, Though the position of kyphosis is significantly less thanthe biconvex core prosthesis. This position is the position of zeroligament stretch (ZLS).

In FIG. 7 the effect of attempting to extend a prosthesis by 10° isshown. Solutions to equations 6, 7 and 8 result in α equalling −1.55°and β equalling 4.75°. The dashed representation of the upper vertebraeagain represents a real disk rotating by 10° about .ACR. This positionrepresents the Lordotic solution to keep points E with the sameco-ordinates. This position represents the 0 ligament stretch position(ZLS).

There are other ways of adding a constraint to the assembly. The otheruseful constraint is to constrain the lower end plate of the uppervertebrae to be parallel with the lower end plate of the upper vertebraein the ‘normal’ situation and to minimize the distance between them.This can be achieved by rotating frames BFR4 and AFR2 by gamma degreesabout the global reference frame AFR1 to produce two new frames AFR3 andBFR5

${{AFR}\; 3} = {{\begin{bmatrix}{\cos \; \gamma} & {{- \sin}\; \gamma} & 0 \\{\sin \; \gamma} & {\cos \; \gamma} & 0 \\0 & 0 & 1\end{bmatrix} \cdot {AFR}}\; 2}$ ${{BFR}\; 5} = {{\begin{bmatrix}{\cos \; \gamma} & {{- \sin}\; \gamma} & 0 \\{\sin \; \gamma} & {\cos \; \gamma} & 0 \\0 & 0 & 1\end{bmatrix} \cdot {BFR}}\; 4}$

For both end plates to be parallel AFR3(1,1)=1(cos(0)=1)  10

and

BFR5(1,3)=AR3(1,3)=0  (11)

as both x coordinates must be the same (zero)

FIG. 8 shows the effect of adding this constraint (constraint 2) to anexisting prosthesis with a biconvex core and attempting to match a 10degree of flexion from a ‘normal’ motion segment. It can be seen thatwith this constraint that the two upper vertebrae cannot superimpose andthat a ligament joining points ACR to E must be stretched beyond itsnormal length. With the constraint that the end plates are parallel,solutions to equation 6, 7 and 8 result in a equalling −1.61° and βequalling −8.39°. The dashed lines represent the real disk rotating by10° about .ACR and the resultant position represents the solution tokeep points with the end plates parallel and with minimum distancebetween them. This constraint is therefore termed Maximal LigamentStretch (MLS). FIG. 9 shows the effect of adding this constraint(constraint 2) to a prostheis with a core with a convex upper surfaceand a concave lower surface and attempting to match a 10 degree offlexion from a ‘normal’ motion segment. It can be seen that with thisconstraint that the two upper vertebrae cannot superimpose and that aligament joining points ACR to E must be stretched beyond its normallength. With the constraint that the end plates are parallel, solutionsto equation 6, 7 and 8 result in α equalling −12.68° and β equalling−2.68°. The dashed lines represent the real disk rotating by 10° about.ACR and the resultant position represents the solution to keep pointswith the end plates parallel and with minimum distance between them.This constraint is therefore termed Maximal Ligament Stretch (MLS).

In FIGS. 6 and 7 the Ligament joining ACR to E has no stretch andinstead the prosthesis rotates at point E to cause a degree of Kyphosisor Lordosis. This constraint is defined a Zero ligament stretch (ZLS)

In the cervical spine there is good anatomical evidence that there isonly a weak posterior longitudinal ligament and the main lateralligaments diverge from near the normal Anatomical centre of rotation(ACR) for that vertebrae. As the anterior longitudinal ligament has, bynecessity, been destroyed by the surgical approach, The main ligamentousconstraint in the cervical spine is approximated by Ligament ACR-E. Inthe absence of an effective posterior longitudinal ligament, there isreason to believe that a cervical disc prosthesis of the type shown inFIG. 6 would behave as if the constraint to movement was that of the ZLSvariety, and there should be a tendency to kyphosis with flexion andlordosis/retrolisthesis in extension.

In the lumbar spine the posterior longitudinal ligament is much tougher.The lumbar spine therefore would preferentially attempt to stretch theligament ACR-E by using the constraint MLS. The annulus fibrosis wouldrarely allow this and the theory would suggest that flexion would belimited.

Whatever the particular case the real constraints in a given disc spacewill be a combination of the constraints ZLS and ALS. The difference inthe angle achieved by the vertebrae and the desired angle(Gamma−(alpha+beta)) in the ZLS situation (Delta A) will be a measure ofthe prostheses inability to match the normal motion required. Thedifference between the length of ligament ACR-E and the desired length(Delta L) will also be a measure of the prostheses inability to matchthe normal motion required.

The mathematical equations developed above will enable design variablesin a 2 articulation prosthesis to be optimized so as to minimize eitherDelta A or Delta L or both. By minimizing Delta A or Delta L theprosthesis will have a better chance of optimally simulating normal.

By the use of simulations using the above mathematical analysis thefollowing holds.

Delta A is minimized to virtually nil by reducing variable Ldsk to zero.This has the effect of moving the prosthetic axis posteriorly so thatthe ACR lies on the Prosthetic Axis. In this position Delta A remainsvery small for all positions of the ACR that lie at or below the diskspace on the prosthetic axis.

Delta A is minimized when the radii of the upper and lower prostheticarticulations are approximately equal.

Delta A is minimized when the radii of the articulations are larger andDelta A gets larger with smaller radii.

It is preferred that Delta A is between 3 and 5°.

The translation of the Core is larger when the Prosthetic Radii arelarger and the translation is smaller when the radii are smaller.

The disclosed prosthesis therefore seeks to. Move the prosthetic axis tothe posterior one third of the disc.

Select optimal radii of the upper and lower joints.

In making these changes two problems are created. In some embodimentsthe core of the prosthesis is no longer symmetrical and was it torotate, it may impinge on the spinal canal.

Because of the posterior positioning of the prosthetic axis the core isat risk of spinal cord impingement.

Based upon the Mathematical Process described above the prosthesisconsists, briefly, of two end plates, an intermediate mobile core and aseparate anterior band for attachment to the upper and lower vertebrae.

FIGS. 13A to 13E show another embodiment of the invention in which theprosthesis consists of a core 50 having concave upper and lower surfaces51, 52. An upper plate 53 has a convex lower surface 54 and lower plate55 has an upper convex surface 56.

The lower surfaces 52, 56 are cylindrical from one side to the other(rotational and translational movement) rather than completelyspherical, whereas the top surfaces 51 and 54 are completely sphericalallowing for universal movement as opposed to backwards and forwardmovement as with the lower surfaces.

An additional feature of the prosthesis 49 shown in these figures is theprovision of upper and lower vertical ridges 57, 58 which are centrallylocated and adapted to fit into grooves created in the bottom surface ofthe upper vertebrae and the upper surface of the lower vertebrae. Asshown more clearly in FIG. 11 the core 50 and upper plate 53 and lowerplate 55 have the prosthetic axis 60 moved to the posterior ⅓ of theprosthesis so that the centre of the upper radius of curvature (CUPR) Aand the centre of the lower radius of curvature (CLPR) D are aligned onthe vertical axis through the ACR. As with the previous embodiment amajor portion 61 is located forward of the axis 60 and a minor portion62 is located behind it. Furthermore, the minor axis of the core 50 isaligned with the vertical axis 61. In addition the anterior andposterior vertical edges of the core 50 are flat and aligned in parallelwith the minor axis 64.

The effect of attempting to flex the prosthesis 49 by 10° with theconstraint being parallel end plates and full ligament stretch resultsin solutions to equations 6, 7 and 8 providing a with an angle of −6.87°and β with an angle of 3.13°.

In FIG. 12 an upper vertebrae 65 rotated through angles α and β arealmost coincident with vertebrae 66 represented in dash line andcorresponding to rotation by 10° (γ) about the ACR. This positionrepresents the solution to keep points with the end plates parallel andwith minimum distance between them. This corresponds to the position ofmaximum ligament stretch (MLS). The core of a bio-concave prosthesis asshown in FIGS. 11 and 12 move anteriorly in flexion. The amount ofligament stretch required to do this is less than when the prostheticaxis is at the mid point of the prosthesis and has therefore a design asshown in FIG. 10A and FIG. 10B. In this configuration the effect ofattempting to flex a prosthesis by 10° with the constraint beingparallel end plates and full ligament stretch results in solutions toequations 6, 7 and 8 providing a with an angle of −6.94° and β with anangle −3.06°. The prosthesis shown in this example represented by item70 is symmetric about its minor axis which also in a state of restcoincides with the vertical axis of the upper and lower vertebrae 71,72. FIG. 10B again shows the effect of moving upper vertebrae 71 throughangles α and β compared to an upper vertebrae rotating by 10° relativeto the ACR. It can be seen that movement possible by upper vertebrae 71does not approximate movement of a real vertebrae 74 as well asprosthesis as designed with a prosthetic axis/minor axis coincident withthe vertical axis through the ACR.

FIG. 14 shows an angled view of the prosthetic device 49 with core 50,upper plate 53 and lower plate 55.

FIGS. 15A and 15B show an alternative embodiment of the invention inwhich a prosthesis is provided with a core 75 with upper plate 76 andlower plate 77. The core 75 has an upper convex surface 78 and a lowerconcave surface 79. As with the embodiments described in relation toFIGS. 12 and 13, the minor axis 80, the prosthetic axis coincides withthe vertical axis through the ACR of the lower vertebrae 81. Because thelower surface 79 is convex it is significantly smaller than the upperconvex surface 78. Likewise the lower surface of the upper plate 76 isconcave and has a matching configuration to surface 78. The lower plate77 has a convex upper surface which is longer than the matching concavesurface 79 to allow movement by the core 75 there over backwards orforwards.

FIG. 15B shows how rotation of the upper vertebrae 82 results inrelative movement between upper plate 76 and core 75 as well as relativemovement between core 75 and lower plate 77.

As with the embodiments shown in FIG. 13 the prosthetic axis isasymmetric and a major portion of the core 75 is located forward of theprosthetic axis.

FIG. 16 shows a side view of another prosthesis 83 consisting of a core84 having an upper convex surface 85 which has a lower radius ofcurvature compared to a lower concave surface 86. In this embodimentboth the upper and lower surfaces 85, 86 have centres of radius ofcurvature which are located below the core 84.

Upper plate 87 has a lower concave surface matching that of surface 85and lower plate 88 has an upper convex surface 89 which is much longerthan the length of the surface 86 to allow reasonable travel backwardsand forwards. In addition the convex surface 89 of the lower plate 88extends into a straight horizontal flat surface 90. This effectivelyprevents forward travel of the core 84 beyond the end of the convexsurface 89.

FIG. 17 shows a prosthesis 91 which is similar to prosthesis 83 exceptthat the upper surface 92 has a greater radius of curvature than thelower surface 93. In addition therefore the lower surface of the upperplate 93 is concave and longer in length than its co acting uppersurface 91. Lower plate 95 has a convex surface which is longer inlength than the co-acting concave surface 92. In addition at a rearwardend of the convex surface 96, an upwardly angled straight section 98 isprovided as a method of stopping movement of the core 99 beyond the endof the convex surface 96.

The forward end of convex surface 96 also extends into a horizontalstraight section 97 which serves to prevent the core 99 moving beyondthe front end of the curved surface 96.

It should be noted that the prostheses 83, 91 are more realisticallyrepresented in FIGS. 16 and 17 as being interposed between upper andlower vertebrae which have a more trapezoidal shape rather than arectangular shape. Thus although surfaces 90 and 97 and previouslydescribed surfaces have been described as being horizontal, in fact theyare slanted and instead are generally parallel to the generalorientation of the upper and lower faces of the upper and lowervertebrae. It should also be noted that surfaces 90 and 97 can be angledupwardly or even downwardly as long as they prevent forward movement ofthe core 84, 99.

The different prosthesis which have been thus far described haveconcentrated on characteristics which emulate an invertebral disk. Anadditional component useful for a prosthesis designed to emulatecharacteristics of an invertebral disk include a band 100 shown in FIG.18 which is designed to closely simulate actions of ligature and in oneembodiment also provides a stop for forward movement of a prostheticcore.

The band 100 consists of a woven fabric 101 consisting of filaments ofwafts and wefts creating a weave with a grid like pattern. Upper andlower ends 102, 103 are provided with connecting plates 104, 105 eachwith holes 106 for screws to be inserted through for attachment to upperand lower vertebrae respectively.

The woven fabric 101 is preferably designed to encourage cellular growthin the interstitial spaces between the threads/filaments and toultimately result in ligatures growing between the upper and lowervertebrae.

According to one embodiment the band is in the form of a prostheticligament made from a woven and absorbable material of appropriatestiffness. The woven material is designed to allow ingrowth of fibroustissue to replace the function of the prosthetic ligament as it isreabsorbed.

According to one embodiment the band is in the form of a gauze made ofwire or polymeric material.

It is preferred that the band is able to elongate or contract in asimilar fashion to a ligament.

With regard to materials used for the different prosthesis describedabove, the end plates may be made from a metal such as titanium,cobalt-chromium steel or a ceramic composite. Typically they have aroughened planar surface which abuts against the adjacent surface of thevertebrae. To assist with fixing the plates to the vertebrae, they maybe provided with a fin or ridge as described in the embodiment shown inFIGS. 13 and 14 or they may be provided with curved surfaces for bearingon an adjacent vertebral body end plate.

The upper and lower surfaces of the core as well as the adjacent curvedsurfaces of the upper and lower end plates are preferably smooth toenhance articulation. The central core may be made from similarmaterials to those used for the end plates, but may also be made from aplastic such as UHMW polyethylene or polyurethane composite.

It is preferred that the radius of curvature of each of the curvedsurfaces of the prosthesis is in the range of 5 to 35 mm.

The foot print of the prosthesis end plates may be of a variety ofshapes but will be optimised to minimise the risk of subsidence into theadjacent vertebral bone.

Although the various articulation surfaces of the core and upper andlower plates have been described in relation to concave and convexsurfaces, it should be noted that other surface profiles are alsoincluded in the invention.

For example the co acting surfaces of the core and the lower plate couldbe ellipsoid instead of cylindrical to provide restricted relativemovement therebetween.

Previously a mathematical explanation has been provided of the behaviourof an artificial disk prosthesis having dual articulation. Differentembodiments of the prosthesis have been described covering each of thepermutations of possible upper and lower surface profiles. These haveincluded biconvex, biconcave as well as convex upper and concave lowerand concave upper and convex lower. The equations previously outlineddescribed the position and orientation of a moving upper vertebrae on afixed lower vertebrae with the dual articulating prosthesis locatedtherebetween. Movement of the upper vertebrae relative to the uppersurface of the prosthesis and movement of the lower surface of theprosthesis relative to the lower vertebrae have been described withreference to constants and by variable angle of rotation of variablesαand β. The orientation of the upper vertebrae is described by:

cos⁻¹(cos α·cos β−sin α·sin β)

The position of a point E immediately above the centre of rotation ofthe disc space and on the lower edge of the upper vertebrae is given bythe following equations:

x(α,β)=−(sin α·cos β+cos α·sin β)·Ldsk+(cos α·cos β−sin α·sinβ)·Cdsk−cos α·Bdsk+Pdsk

y(α,β)=−(sin α·cos β+cos α·sin β)·Ldsk+(cos α·cos β−sin α·sinβ)·Cdsk−cos α·Bdsk+Pdsk

Where Constants define the size and functional type of the prosthesis.

Depending on the relative sizes of parameters Ldsk, Cdsk, Bdsk and Pdskthere are 4 distinct types of prosthesis that are described:

These are: (described by the core shape)

1. Biconvex

2. Biconcave

3. Convex top concave bottom with the top radius greater than bottomradius

4. Convex top concave bottom with the bottom radius greater than the topradius

Equations 1-3 describe the kinematics of these 4 prosthesis.

The length of a line joining the COR to point E is

l(α,β)=√{square root over (x(α,β)² +y(α,β)²)}{square root over (x(α,β)²+y(α,β)²)}

Wherein α is the angular displacement of the upper part relative to themiddle part, β is the angular displacement of the middle part relativeto the lower part and l is the ligature joining a part of the upper partwith the centre of rotation of the skeletal structure (or prosthesis)when in use and where x(α,β) and y(α,β) are different functions.

Preferably “ligature” includes any elongate member particularly one witha degree of extension of stretch and contraction or compression.

The values of alpha and beta can be calculated that produce a minimumvalue for l. l can be considered to be the lateral ligament of thespinal motion segment. As this is elastic it can be seen that it willbehave as a spring and consequently will have the lowest elasticpotential energy when l is smallest. An equilibrium position can becalculated when l is either a minimum or a maximum. Mathematically thiscan be defined as the gradient vector being zero:

$\begin{matrix}{{\nabla{l( {\alpha,b} )}} = \begin{bmatrix}\frac{\delta \; l}{\delta \; \alpha} \\\frac{\delta \; l}{\delta \; \beta}\end{bmatrix}} \\{= \begin{bmatrix}0 \\0\end{bmatrix}}\end{matrix}$

Under the circumstances of a zero gradient vector the prosthesis willhave a zero change in elastic potential energy for infinitesimal changesin α and β and the prosthesis will be in an equilibrium position.However when this equilibrium position is a maximum value for l it canbe seen that small perturbations in α or β will tend to cause l todecrease and the equilibrium position is unstable. Mathematically thiscan be described as:

$\begin{bmatrix}\frac{{\delta \;}^{2}l}{\delta \; \alpha} \\\frac{{\delta \;}^{2}l}{\delta \; \beta}\end{bmatrix} = \begin{bmatrix}{- {ve}} \\{- {ve}}\end{bmatrix}$ ${{or}\begin{bmatrix}\frac{{\delta \;}^{2}l}{\delta \; \alpha} \\\frac{{\delta \;}^{2}l}{\delta \; \beta}\end{bmatrix}} = \begin{bmatrix}{+ {ve}} \\{- {ve}}\end{bmatrix}$ ${{or}\begin{bmatrix}\frac{{\delta \;}^{2}l}{\delta \; \alpha} \\\frac{{\delta \;}^{2}l}{\delta \; \beta}\end{bmatrix}} = \begin{bmatrix}{- {ve}} \\{+ {ve}}\end{bmatrix}$

Under these circumstances it would be possible for the prosthesis to beprecisely balanced and be in equilibrium, though small perturbationswould cause it to rapidly adopt a position of maximum flexion orextension.

It can be shown that prostheses 1) and 4) have unstable equilibriumpositions. This situation occurs when matching an on axis or off axisCOR. By increasing the radii of the upper and lower articulations thevalue of the gradient vector will be less negative and the tendency toadopt a position of maximum flexion or extension will be diminished.

Prostheses 2) and 3) however, have the property of having a positivesecond partial derivative of l.

$\begin{bmatrix}\frac{{\delta \;}^{2}l}{\delta \; \alpha} \\\frac{{\delta \;}^{2}l}{\delta \; \beta}\end{bmatrix} = \begin{bmatrix}{+ {ve}} \\{+ {ve}}\end{bmatrix}$

At the equilibrium position defined by

$\begin{matrix}{{\nabla{l( {\alpha,\beta} )}} = \begin{bmatrix}\frac{\delta \; l}{\delta \; \alpha} \\\frac{\delta \; l}{\delta \; \beta}\end{bmatrix}} \\{= {\begin{bmatrix}0 \\0\end{bmatrix}.}}\end{matrix}$

In other words there exist values of α and β that cause an equilibriumwhen l is a minimum. This equilibrium is stable and self correcting orself centering because any tendency to perturb the prosthesis from theequilibrium position will have a tendency to cause l to lengthen andtherefore increase the elastic potential energy of the system.

Prostheses 2) and 3) therefore have stable equilibrium positions and areself correcting or self centering.

When attempting to match a COR that is on axis with the prosthesis theequilibrium position is in the neutral position (when α=β=0). Whenmatching an off axis COR the equilibrium position will now be locatedaway from the neutral position (when α≠β≠0)—when matching a CR that isanterior to the prosthesis axis the equilibrium position will be inextension and when matching a CR that is posterior to the prostheticaxis the equilibrium position will be in flexion.

When the values of the radii of the upper and lower articulations arelarge the equilibrium position changes due to matching an offset will belessened.

FIG. 20A shows a graphical plot in three dimensions of the ligamentlength l versus β and α for a prosthesis core with a convex uppersurface and a concave lower surface where the radius of curvature of theupper surface of greater than that of the lower surface. As an examplethe upper radius is 36 mm compared to 12 mm for the lower radius.

It can be seen from FIG. 20A that the three dimensional graph shownindicates a minimum ligament length as represented by a trough in thegraph.

In this Figure there is zero, X and Y offset, which means the prosthesisaxis is aligned with the patient's centre of rotation and corresponds toL in FIGS. 19A and 19B.

FIG. 20B shows the effect of introducing a value for L of 1 mm for thesame type of prosthesis shown in FIG. 20A. The equilibrium positionmoves in the opposite direction to L. The mathematical method can beused to optimise this change in equilibrium position and make theprosthesis less sensitive to changes in Y offset (L) or X offset (movingthe patient's CR inferiorly).

FIG. 21 shows a two dimensional view of the change in ligament lengthwith angle of flexion for the prosthesis referred to in FIG. 20A. Inthis Figure it can be seen that there is a clear trough around thecentre of rotation represented by angle of flexion extension 0. Thisgraph clearly shows that any movement of the prosthesis away from thecentre of rotation results in extension of the ligament length andtherefore a natural tendency for the ligament to want to return to itsminimum length at the centre of rotation.

The above contrasts with a prosthesis with a biconvex core. A graphicalsolution to the mathematical equation outlined above is shown in FIG. 22for a biconvex core. It should be apparent from this graphical analysisthat there is no minimum ligament length which provides a point ofequilibrium. In fact the two dimensional graphical representation shownin FIG. 23 shows how the point of equilibrium is located about thecentre of rotation of the biconvex core and shows that any movement ofthe core away from the centre of rotation results in a decrease in theligaments length and therefore a tendency for the core to move away fromthe point of equilibrium.

FIG. 24A shows another embodiment of the invention in which theprosthesis has a biconcave core. As with the embodiment shown in FIGS.20A, 20B and 21 in this embodiment there is a point of equilibrium aboutthe centre of rotation for the core. This point of equilibriumcorresponds to the minimum ligature length and hence provides a naturaltendency for the core to return to the point of equilibrium if there ismovement away from the centre of rotation.

FIG. 24B shows the variation in ligament length for a biconcaveprosthesis with the upper radius equal to 36 mm and the lower radiusequal to 36 mm with Y offset and X offset being zero. It can thus beseen that with equal radius the graphical representation of themathematical model shows there is no tendency for movement of theprosthesis away from the equilibrium position to result in a movementback to the equilibrium position.

The two dimensional graphical representation which is not shown has asimilar appearance to FIG. 21.

From the above it should be clear that if it is desired to produce aprosthesis with a self correcting ability which results in a tendency ofany movement away from a point of equilibrium to result in a naturalurging of the prosthesis back towards the point of equilibrium, then theembodiments of the invention described in relation to FIGS. 20A, 20B, 21and 24A provide suitable solutions. It is also noted however that theother embodiments of the invention which have been described may stillbe used even though they may not have this self-centering ability. Thisis because other alterations may be made to the overall prosthesis inorder to keep movement of the prosthesis within predeterminedboundaries.

All prostheses are able to match a COR that is on axis. When the COR isoff axis they can only match by either stretching or shortening thelateral ligament (Delta L) or by adopting abnormal orientation (DeltaA). Delta A and Delta L, for a given offset can be reduced by making theRadii Larger.

All prostheses are capable of pure translation. Prosthesis 2) does sowith loss of disc height. Prosthesis 3) does so with a gain in discheight.

The Ideal prosthesis has

1 Stable equilibrium position

2 The best capacity to handle off axis CR

This is 2) or 3)

The preferred embodiment is 3) with as large radii as possible. This hasthe added advantage of resistance to pure translation (because the softtissues would need to lengthen)

The second preferred embodiment is 2) this has the advantage ofrelatively unrestricted translation.

There may be clinical situations where either 2) or 3) are preferable.

The ratio of radii for prosthesis 3 can be set so that under a COR matchto a physiologically normal centre of rotation there is equal arc travelbetween the top and bottom articulation. This can be achieved by

R1*alpha=R2*beta

Where alpha and beta can be calculated for the CR match.

This will match wear for upper and lower articulations and is the mostefficient use of the surface area of contact.

A capacity to calculate alpha and beta allows the prosthesis to bedrafted.

An ideal range for of ratio for prostheses in the lumbar spine andcervical spines to achieve this is 3:1-10:1.

In the lumbar and cervical spines the preferred radii are 5 mm and 50mm.

Prosthesis 2 does not allow the travel on each articulating surface tobe equal. Preferably the mathematical model will be used for thisprosthesis to allow an optimum choice of ratio of radii based on anydesirable parameter such as the desired ratio of angles α and β. In thelumbar and cervical spines the ideal ratio is 2:1-1:2. In the Lumbarspine the preferred radii are between 8-40 mm for each radii. In thecervical spine the preferred radii are between 6-30 mm

FIGS. 10A to 11B and 13A to 13F show a prosthesis having a biconcaveprofile. FIGS. 15A, 15D, 16 and 17 show a prosthesis with an upperconvex surface and a lower concave surface. It is preferred that theversion of the prosthesis shown in FIG. 17 is utilised as the radius ofcurvature of the upper surface is larger than that of the lower concavesurface. It is also preferred that the lower surface of the upper platehas a matching profile to the upper surface of the prosthesis and theupper surface of the lower plate has a matching profile to the lowersurface of the prosthesis. It should be understood however that thisdoes not mean that the entire lower surface of the upper plate and uppersurface of the lower plate have the matching profiles. Thus a referenceto FIG. 19A shows one preferred configuration of a prosthesis having thepreferred upper and lower surface profiles identified above. Theprosthesis 110 shown is symmetrical about a central vertical axis andhas smooth curved outer upper and lower edges. The prosthesis is shownoffset rearwardly and hence with its core 113 retained within a rearwardregion of the upper and lower plates 111, 112.

The upper surface of the lower plate 112 has a rearward portion having aconvex shape of matching configuration to the opposing concave profileof the core 113.

The apex of the convex region is offset rearwardly with respect to thecentre of prosthesis. The convex region is symmetrical about its offsetcentral vertical axis and on either side extends into a concave troughwith the result that the overall profile of this region has theappearance of part of a sinusoidal curve.

Each of the troughs extend into upwardly curved surfaces on either sideof the convex region and provide rearward and forward detents to limitforward and backward movement of the prosthesis relative to the lowerplate.

The upper plate 111 has a rearwardly offset concave lower surface with arearmost downwardly extending edge which is configured to limit rearwardmovement of the core 113 relative to the upper plate 111.

As can be seen from FIG. 19A the radius of curvature of the lowersurface of the upper plate is larger than the radius of curvature of theupper surface of the lower plate. In each case the radius of curvaturehas a common origin which is located on the central vertical axis of theprosthesis at a virtual point below the prosthesis.

FIG. 19A also shows the core 113 in a stable equilibrium positionaligned with the central vertical axis 114 which is rearwardly offset tothe anatomical central axis.

A lateral ligament 115 is shown connected between upper and lowervertebrae 116, 117. The ligament 115 is offset by a distance L from theaxis 114.

FIG. 19B shows how the prosthesis 112 is effected by backward rotationalmovement of the upper vertebrae 116 with respect to the lower vertebraldisk 117.

As shown the lateral ligament 115 rotates about the centre of rotation(CR) and the upper plate 111 and core 113 rotate to an unstableposition. Because of the difference in radius of curvature of the uppersurface of core 113 and the lower surface the ligament 115 is stretchedand there is a natural tendency for it to return to the equilibriumposition shown in FIG. 19A. The difference in radius of curvature alsomeans that the upper vertebral disk 116 will rotate and translate withrespect to the lower vertebral disk 117.

By increasing the radius of curvature of the lower surface of the upperplate and hence the upper surface of the prosthesis it is possible toincrease the stability of the prosthesis when in use if the COR of theprosthesis is offset from the ACR.

Desirably increasing the radius of curvature of the upper surface of theprosthesis and hence the lower surface of the upper plate enhances theease with which the prosthesis is able to return to its equilibriumpoint centered about its central vertical axis.

According to one embodiment the further the central vertical axis of theprothesis is offset from the centre of rotation of the skeletalstructure, the larger the radius of curvature of the upper surface ofthe prosthesis.

It is preferred that the radius of curvature of the upper surface of theprosthesis is between 30 and 50 mm in the lumbar spine and 20 and 40 mmin the cervical spine. As it is preferred that the ratio of the radiusof curvature of the upper surface of the prosthesis compared to thelower surface of the prosthesis is within a predetermined range anincrease in the radius of curvature of the upper surface of theprosthesis will result in a corresponding increase in radius ofcurvature of the lower surface of the prosthesis.

It is preferred that the length of the convex region of the uppersurface of the lower plate (when measured from front to rear) isdetermined in accordance with typical travel allowed for a vertebraldisk in a typical vertebral column.

It is preferred that all embodiments of the prostheses have a prostheticaxis that is set in the posterior h of the disc to as closely aspossible match the normal physiological centre of rotation.

It is to be understood that, if any prior art publication is referred toherein, such reference does not constitute an admission that thepublication forms a part of the common general knowledge in the art, inAustralia or in any other country.

1. A prosthesis for a vertebral column comprising an upper part forattachment to an upper vertebrae, a lower part for attachment to a lowervertebrae and a middle part located between the upper and lower parts,wherein the upper part has a lower surface portion with a first radiusof curvature, the middle part has an upper surface portion with a secondradius of curvature and a lower surface portion with a third radius ofcurvature and the lower part has an upper surface portion with a fourthradius of curvature, wherein the center of the radius of curvature forat least two surfaces is offset rearwardly with respect to a centralvertical axis through the upper and lower vertebrae and/or the upper andlower parts.
 2. The prosthesis as claimed in claim 1 wherein at leastone of the fourth radius of curvature and the first radius of curvatureis offset rearwardly of the central vertical axis.
 3. The apparatus asclaimed in claim 1 wherein the center of radius of curvature of each ofthe surfaces is offset rearwardly with respect to the central verticalaxis.
 4. The prosthesis as claimed in claim 1 wherein the center of theradius of curvature for each of the surfaces is located in the posteriorthird of the prosthesis.
 5. The prosthesis as claimed in claim 1 whereinthe middle part has a minor central axis and a major central axis, theminor central axis being located through the center of the radius ofcurvature of the second and third surfaces.
 6. The prosthesis as claimedin claim 5 wherein the minor central axis is inclined with respect tothe vertical central axis.
 7. The prosthesis as claimed in claim 5wherein the major axis is located through the center of the posteriorand anterior ends of the middle part.
 8. The apparatus as claimed inclaim 1 wherein the middle part has a convex upper surface and a concavelower surface.
 9. The prosthesis as claimed in claim 1 wherein the uppersurface of the middle part is concave and the lower surface of themiddle part is concave.
 10. The prosthesis as claimed in claim 8 whereinthe radius of curvature of the upper surface of the middle part isgreater than the radius of curvature of the lower surface.
 11. Theprosthesis as claimed in claim 8 wherein the lower surface first radiusof curvature is substantially the same as the radius of curvature of thesecond radius of curvature.
 12. The prosthesis as claimed in claim 8wherein the third radius of curvature is substantially the same as thefourth radius of curvature.
 13. The prosthesis as claimed in claim 1wherein the first, second, third and fourth radius of curvature arecentered on a vertical axis rearwardly offset from the central verticalaxis through the upper part and lower part.
 14. The prosthesis asclaimed in claim 1 wherein the upper part has a lower surface frontportion which is substantially flat.
 15. The prosthesis as claimed inclaim 1 wherein the lower part has an upper surface front portion whichis substantially flat.
 16. The prosthesis as claimed in claim 1 whereinthe lower part has an upper surface comprising a rear surface portionand a front surface portion, wherein the rear surface portion comprisesa convex portion symmetrical about a vertical central axis through themiddle part.
 17. The prosthesis as claimed in claim 16 wherein the uppersurface rear portion of the lower part comprises recessed portions oneither side of the convex portion.
 18. The prosthesis as claimed inclaim 1 wherein the middle part has an upper convex surface and a lowerconvex surface with the radius of curvature of each of the upper andlower surfaces of the middle part being configured whereby movement ofthe middle part relative to the upper part or lower part, within apredetermined range results in the middle part being urged to a positionof equilibrium located substantially along a central axis of theprosthesis.
 19. The prosthesis as claimed in claim 1 wherein the middlepart has an upper concave surface and a lower concave surface with theradius of curvature of each of the upper and lower surfaces of themiddle part being configured whereby movement of the middle partrelative to the upper part or lower part, within a predetermined rangeresults in the middle part being urged to a position of equilibriumlocated substantially along a central axis of the prosthesis.
 20. Theprosthesis as claimed in claim 18 wherein the second radius of curvatureis greater than the third radius of curvature.
 21. The prosthesis asclaimed in claim 20 wherein the ratio of the second radius of curvaturecompared to the third radius of curvature is between 3:1 to 10:1. 22.(canceled)
 23. The prosthesis as claimed in claim 1 wherein the centerof rotation of the upper part, middle part and lower part is rearwardlyoffset with respect to a central vertical axis through the prosthesis.24. The prosthesis as claimed in claim 23 wherein the radius ofcurvature of each surface is configured whereby movement of any one ormore parts within a predetermined range results in the middle part beingurged to a position of equilibrium located substantially along a centralvertical axis of the prosthesis.
 25. The prosthesis as claimed in claim1 wherein the prosthesis has a center of rotation corresponding to aposition of equilibrium for the middle part.
 26. The prosthesis asclaimed in claim 1 wherein a major portion of the middle part isconfigured to be located forward of the anatomical center of rotation ofa vertebral column when the upper and lower vertebrae are substantiallyvertically aligned.
 27. The prosthesis as claimed in claim 1 wherein theprosthesis includes a stopping means located behind and in front of themiddle part when the prosthesis is located in a vertebral column. 28.The prosthesis as claimed in claim 27 wherein the stopping meansincludes end portions of the upper and lower parts.
 29. The prosthesisas claimed in claim 1 wherein the middle part has a position ofequilibrium defined by: $\frac{\delta \; l}{\delta \; \alpha} = o$$\frac{\delta \; l}{\delta \; \beta} = o$ and$\frac{\delta^{2}\; l}{\delta \; \beta^{2}} = {+ {ve}}$ and$\frac{\delta^{2}\; l}{\delta \; \beta^{2}} = {+ {ve}}$ wherein α isthe angular displacement of the upper part relative to the middle part βis the angular displacement of the middle part relative to the lowerpart; and l is the length of a ligature joining a part of the upper partwith the center of rotation of the skeletal structure when in use andwherein:l(α,β)=√{square root over (x(α,β)² +y(α,β)²)}{square root over (x(α,β)²+y(α,β)²)} where x(α,β) and y(α,β) are different functions.
 30. Theprosthesis as claimed in claim 1 wherein the middle part is configuredwith an upper surface and a lower surface with a position of equilibriumrelative to the upper and lower parts being defined by:$\frac{\delta \; l}{\delta \; \alpha} = o$$\frac{\delta \; l}{\delta \; \beta} = o$ and$\frac{\delta^{2}\; l}{\delta \; \beta^{2}} = {+ {ve}}$ and$\frac{\delta^{2}\; l}{\delta \; \beta^{2}} = {+ {ve}}$ wherein α isthe angular displacement of the upper part relative to the middle part βis the angular displacement of the middle part relative to the lowerpart; and l is the length of a ligature joining a part of the upper partwith the center of rotation of the skeletal structure when in use andwherein:l(α,β)=√{square root over (x(α,β)² +y(α,β)²)}{square root over (x(α,β)²+y(α,β)²)} where x(α,β) and y(α,β) are different functions; whereinsmall changes in the length l relative to small changes in either theangle α or β is a minimum and either side of the position of equilibriumthe length l increases if the middle part moves away from the positionof equilibrium.
 31. The prosthesis as claimed in claim 30 wherein eachof the variables α, β or l are determined relative to the center ofrotation of the prosthesis.
 32. The prosthesis as claimed in claim 1wherein the upper part, middle part and lower part are configuredwhereby the prosthesis has a position of equilibrium aligned with acentral vertical axis offset from an anatomical central axis and isconfigured with a self-centering means which tends to cause theprosthesis to align with the equilibrium position if there is movementof any one of the parts to move away from the position of equilibrium.33. The prosthesis as claimed in claim 32 wherein the self-centeringmeans comprises a predetermined surface configuration for the first,second, third and fourth radius of curvature respectively which resultsin a tendency for the prosthesis to self-center about the position ofequilibrium whenever the upper part and/or middle part move away fromthe position of equilibrium.
 34. The prosthesis as claimed in claim 1wherein the upper part, middle and lower part have a stable equilibriumposition.
 35. The prosthesis as claimed in claim 34 wherein theequilibrium position is offset with respect to a vertical central axisthrough the prosthesis.
 36. The prosthesis as claimed in claim 34wherein the first, second, third and fourth radius of curvature areconfigured to urge the prosthesis to the stable equilibrium position ifthere is relative movement between any of the upper part, middle partand lower part.
 37. The prosthesis as claimed in claim 1 wherein theupper part when the prosthesis is attached to upper and lower vertebrae,closely simulates rotational and translational movements possible withan invertebral disk. 38-57. (canceled)